May 16
Ali Enayat,
University of Gothenburg
Tarski's undefinability of truth theorem strikes again: part II
Tarski's undefinability of truth theorem has two versions, the first one deals with truth itself, takes some effort to prove, and is a descendant of the Epimenides (liar) paradox. The second one deals with the related concept of satisfaction, has a one-line proof, and is a descendent of Russell's paradox. This talk is about the first one, which appeared in the 1953 monograph 'Undecidable Theories' by Tarski, Mostowski, and Robinson; it was employed there to show the essential undecidability of consistent theories that can represent all recursive functions (a strong form of the Gödel-Rosser incompleteness theorem). I will present Tarski's original 1953 formulation (which differs from the common formulation in modern expositions) and will explain how it was used in my recent work with Albert Visser to show that no consistent completion of a sequential theory whose signature is finite is axiomatizable by a collection of sentences of bounded quantifier-alternation-depth. A variant of this result was proved independently by Emil Jeřábek, as I will explain. Our proof method has a pedagogical dividend since it allows one to replace the cryptic Gödel-Carnap fixed point lemma with the perspicuous undefinability of truth theorem in the proof of the Gödel-Rosser incompleteness theorem.